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The table maker's quantum search

Stefanos Kourtis

TL;DR

The paper addresses the problem of determining the hardness to round $\mathrm{htr}_{f,I}(n)$ for elementary functions, a formulation of the table maker's dilemma. It introduces a quantum algorithm that uses Grover search with a tailored membership oracle to identify the minimal working precision or an upper bound, by binary-searching over the working precision $p$ and evaluating $f(x)$ in coherent superposition over $n$-bit inputs. For functions related to the exponential, and under a polynomial upper bound $p_{\max}=O(\mathrm{poly}(n))$, the method achieves a quantum runtime of $\tilde{O}(2^{n/2}\log(1/\delta))$, representing a modest asymptotic speedup over best-known classical approaches in large binades for periodic functions. The work demonstrates, for the first time, an asymptotic quantum speedup in a computer arithmetic task and discusses practical considerations, limitations, and open directions for combining quantum and classical techniques to estimate hardness to round.

Abstract

We show that quantum search can be used to compute the hardness to round an elementary function, that is, to determine the minimum working precision required to compute the values of an elementary function correctly rounded to a target precision of $n$ digits for all possible precision-$n$ floating-point inputs in a given interval. For elementary functions $f$ related to the exponential function, quantum search takes time $\tilde O(2^{n/2} \log (1/δ))$ to return, with probability $1-δ$, the hardness to round $f$ over all $n$-bit floating-point inputs in a given binade. For periodic elementary functions in large binades, standalone quantum search yields an asymptotic speedup over the best known classical algorithms and heuristics.

The table maker's quantum search

TL;DR

The paper addresses the problem of determining the hardness to round for elementary functions, a formulation of the table maker's dilemma. It introduces a quantum algorithm that uses Grover search with a tailored membership oracle to identify the minimal working precision or an upper bound, by binary-searching over the working precision and evaluating in coherent superposition over -bit inputs. For functions related to the exponential, and under a polynomial upper bound , the method achieves a quantum runtime of , representing a modest asymptotic speedup over best-known classical approaches in large binades for periodic functions. The work demonstrates, for the first time, an asymptotic quantum speedup in a computer arithmetic task and discusses practical considerations, limitations, and open directions for combining quantum and classical techniques to estimate hardness to round.

Abstract

We show that quantum search can be used to compute the hardness to round an elementary function, that is, to determine the minimum working precision required to compute the values of an elementary function correctly rounded to a target precision of digits for all possible precision- floating-point inputs in a given interval. For elementary functions related to the exponential function, quantum search takes time to return, with probability , the hardness to round over all -bit floating-point inputs in a given binade. For periodic elementary functions in large binades, standalone quantum search yields an asymptotic speedup over the best known classical algorithms and heuristics.
Paper Structure (6 sections, 1 theorem, 11 equations, 1 algorithm)

This paper contains 6 sections, 1 theorem, 11 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Preliminaries and notation
  4. Grover search and membership oracle
  5. Proof of Theorem \ref{['theorem:htr']}
  6. Discussion

Key Result

Theorem 1

For any elementary function $f$ evaluated at precision-$n$ binary floating-point inputs $x\in I$ where $I=[2^e,2^{e+1})$, Algorithm alg:qhtr returns $l = \min(\{ \mathrm{htr}_{f,I}(n), p_{\max} \})$ with probability $1-\delta$ by making $O(2^{n/2} \log\frac{\lfloor \log p_{\max} \rfloor + 1}{\delta}

Theorems & Definitions (1)

  • Theorem 1